All People in Canada are the Same Age

Date: 01/18/2002 at 10:01:52
From: Kathleen Webb
Subject: Inductive proof
Here is an "absurd" proof where I must show the error in the proof.
I'm having problems understanding where the logic breaks down, even
though I'm sure it's absurd.
"For n = 1,2,3,..., every bag contains n solid-colored balls of only
one color."
Prove for n = 1: A bag with one ball clearly has balls of only one
color.
Prove for n+1: Take a bag with n+1 balls. Remove one ball. By
hypothesis, the bag now has balls of only one color. Replace the
first ball and remove a different ball. Again, the bag now has balls
of only one color. Since the other balls that never left the bag
did not change color, all the balls must be the same color.
Can you help me find the error in this inductive proof?
Thanks!

Date: 01/18/2002 at 12:20:15
From: Doctor Paul
Subject: Re: Inductive proof
This is a famous problem that can be rephrased in many different way.
The same argument can show that all horses are the same color and that
everyone in Canada is the same age and so forth...
You'll find the breakdown in the proof here:
All People in Canada are the Same Age - Univ. of Toronto
http://www.math.toronto.edu/mathnet/falseProofs/sameAge.html
Be warned - it's *very* subtle. I think it's kind of neat to see how
close you can come to proving such a patently false statement. :-)
I hope this helps. Please write back if you'd like to talk about this
more.
- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/